1. What's the best strategy to follow in building a team on a limited budget?
There are three key draft strategies in most every league:
A) Figure out how many IP and PA to draft in order to maximize performance
B) Read the theme rules regarding playoff structure, trans deadline and fatigue. Decide how many games it will take to win and split your squad accordingly.
C) Draft with an eye for bargains players that perform well relative to their $/ip or $/pa.

2. Will the below average AAA players contribute at all, or will they just provide PA and IP as filler? What regular player salary range are they comparable to?
Likely one or two will have deeper usefulness, but I would only plan on them being filler IP and PA.

3. On the pitching side, going with 5 starters in the 200 Innings range, two relievers in the 115 innings range, and a closer. For all 8 of these, use whips in 1.3 range, except even go to 1.4 if the guy keeps the ball in the park (HR/9 around .5). Is it smart to sacrifice some on the whip to get the lower HR/?
A) As TJ wrote don't worry about the innings each player has. Draft with an eye to total innings and front-end quality.
B) At this cap and with this AAA, I would expect deadballers to dominate.

4. The ninth staff guy would be a basic low salary stiff to serve as mop up...That should get me to about 1350-1400 innings total. Do I need to draft another decent reliever or will the AAA guys be effective enough?
As TJ wrote 1350 drafted innings is probably too much.

5. For the hitters...concentrate on defense at second, third, short, and CF...but one question I have is: in a low budget league like this should I settle for B fielders at these positions for the sake of better offense?
I think the most cost effective players at $60M are worse defenders than the most cost effective players at $80M. So your defense probably would be worse at $60M, but I can't say that you'd have to go with B fielders.

6. At LF, RF, and 1B, worry little about defense...what I'm thinking about for offense at these positions is a platoon....since I can't afford one high budget player at these positions concentrate at two guys at each position with 350 PA who are darn good in their use and then platooning them on a "vs. lefty-righty" basis...
A) 1st, LF, RF and to an extent 3B are positions where it is likely worthwhile to sacrifice defense for offense.
B) With due respect to TJ, I'm not a huge fan of platoons as it is generally hard to find room on your roster if you maximize your AAA and bargains.